The angles of the triangle are 81.870°, 81.870° and 16.260°, respectively.
We can determine the angles of the triangle by using the definition of dot product as follows:
[tex]\alpha = \cos^{-1} \frac{\overrightarrow {QP}\,\bullet \,\overrightarrow {QR} }{\|QP\|\cdot \|QR\|}[/tex] (1)
[tex]\beta = \cos^{-1} \frac{\overrightarrow {RQ}\,\bullet \,\overrightarrow {RP} }{\|RQ\|\cdot \|RP\|}[/tex] (2)
[tex]\gamma = \cos^{-1} \frac{\overrightarrow{PR}\,\bullet\,\overrightarrow {PQ}}{\|\overrightarrow {PR}\|\cdot \|\overrightarrow{PQ}\|}[/tex] (3)
Where:
- [tex]\alpha[/tex], [tex]\beta[/tex], [tex]\gamma[/tex] - Angles of the triangle.
- [tex]\overrightarrow {PQ}[/tex], [tex]\overrightarrow {QR}[/tex], [tex]\overrightarrow {PR}[/tex] - Vectors of the triangle.
- [tex]\|\overrightarrow{PQ}\|[/tex], [tex]\|\overrightarrow{QR}\|[/tex], [tex]\|\overrightarrow{PR}\|[/tex] - Norm of the vectors of the triangle.
Please remember that Norm is determined by the Pythagorean Theorem.
If we know that [tex]P(x,y) = (1,0)[/tex], [tex]Q (x,y) = (0,1)[/tex] and [tex]R(x,y) = (4, 4)[/tex], then the angles of the triangle are, respectively:
[tex]\overrightarrow {PQ} = (0,1)-(1,0)[/tex]
[tex]\overrightarrow{PQ} = (-1, 1)[/tex]
[tex]\overrightarrow{QP} = (1, -1)[/tex]
[tex]\|\overrightarrow {PQ}\| = \|\overrightarrow{QP}\| = \sqrt{2}[/tex]
[tex]\overrightarrow {QR} = (4,4) - (0,1)[/tex]
[tex]\overrightarrow {QR} = (4, 3)[/tex]
[tex]\overrightarrow {RQ} = (-4, -3)[/tex]
[tex]\|\overrightarrow{QR}\| = \|\overrightarrow {RQ}\| = 5[/tex]
[tex]\overrightarrow {PR} = (4,4) - (1,0)[/tex]
[tex]\overrightarrow {PR} = (3,4)[/tex]
[tex]\overrightarrow{RP} = (-3,-4)[/tex]
[tex]\|\overrightarrow {PR}\| = \|\overrightarrow{RP}\| = 5[/tex]
[tex]\alpha = \cos^{-1} \frac{(1,-1)\,\bullet (4,3)}{5\sqrt{2}}[/tex]
[tex]\alpha \approx 81.870^{\circ}[/tex]
[tex]\beta = \cos^{-1} \frac{(-4,-3)\,\bullet\,(-3, -4)}{25}[/tex]
[tex]\beta \approx 16.260^{\circ}[/tex]
[tex]\gamma = \cos^{-1} \frac{(3,4)\,\bullet\,(-1, 1)}{5\sqrt{2}}[/tex]
[tex]\gamma \approx 81.870^{\circ}[/tex]
The angles of the triangle are 81.870°, 81.870° and 16.260°, respectively.
We kindly invite to check this question on triangles: https://brainly.com/question/15161714
